Vol. 12, No. 5, 2018

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Polynomial bound for the nilpotency index of finitely generated nil algebras

Mátyás Domokos

Vol. 12 (2018), No. 5, 1233–1242
Abstract

Working over an infinite field of positive characteristic, an upper bound is given for the nilpotency index of a finitely generated nil algebra of bounded nil index $n$ in terms of the maximal degree in a minimal homogenous generating system of the ring of simultaneous conjugation invariants of tuples of $n$-by-$n$ matrices. This is deduced from a result of Zubkov. As a consequence, a recent degree bound due to Derksen and Makam for the generators of the ring of matrix invariants yields an upper bound for the nilpotency index of a finitely generated nil algebra that is polynomial in the number of generators and the nil index. Furthermore, a characteristic free treatment is given to Kuzmin’s lower bound for the nilpotency index.

Keywords
nil algebra, nilpotent algebra, matrix invariant, degree bound
Mathematical Subject Classification 2010
Primary: 16R10
Secondary: 13A50, 15A72, 16R30